Cosmology and tests of General Relativity with gravitational waves: latest results from LIGO-Virgo, and outlook.

In this talk I will explain how current, and also future, GW observations can help probe the standard cosmological model. In the first part I will focus on the determination of the Hubble Constant with LIGO-Virgo, providing the latest results, and explaining the orgin of different sources of errors in the Hubble constant measurement. Then we will discuss what GWs can say about what lies beyond the standard Lambda CDM cosmological model, namely how with GWs one can probe theories of gravity beyond general relativity.